Question: The Smiths and the Johnsons were competing in the final leg of the Amazing Race. In their race to the finish, the Smiths immediately took off on a $165$ kilometer path traveling at an average speed of $v$ kilometers per hour. The Johnsons' start was delayed by $\dfrac{1}{2}$ hour. Eventually, they took a $180$ kilometer path to the finish, traveling at an average speed that was $20$ kilometers per hour faster than the Smiths' speed. The Johnsons arrived at the finish line first and won the race! Write an inequality in terms of $v$ that models the situation.
Answer: The strategy In the final leg of the Amazing Race, the team that arrives first wins the race. Since the Johnsons win the race, it takes them less time to complete the race than the Smiths. If we let $J$ be the time it takes for the Johnsons to complete the race and $S$ be the time it takes for the Smiths to complete the race, we have that $J<S$. Now, let's express $J$ and $S$ in terms of $v$. Expressing the time it takes for the Smiths to complete the race We know that $\text{distance}=\text{speed}\cdot \text{time}$ and so $\text{time}=\dfrac{\text{distance}}{\text{speed}}$. Since the Smiths traveled $165$ kilometers at an average speed of $v$ kilometers per hour, it took them $\dfrac{165}{v}$ hours to complete the race. Expressing the time it takes for the Johnsons to complete the race We know that the Johnsons' average speed was $20$ kilometers per hour faster than the Smiths' average speed, or $v+20$ kilometers per hour. Since the Johnsons traveled $180$ kilometers at an average speed of $v+20$ kilometers per hour, it took them $\dfrac{180}{v+20}$ hours to complete the race. However, they were delayed by $\dfrac{1}{2}$ hour, so their total time was $\dfrac{180}{v+20}+\dfrac12$ hours. Putting things together We found that $S=\dfrac{165}{v}$ and $J=\dfrac{180}{v+20}+\dfrac12$. Since $J<S$, we can substitute and find an inequality in terms of $v$ that models the situation. The answer is: $ \dfrac{180}{v+20}+\dfrac12<\dfrac{165}{v}$